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Ufa Mathematical Journal, 2020, Volume 12, Issue 1, Pages 103–113
DOI: https://doi.org/10.13108/2020-12-1-103
(Mi ufa506)
 

This article is cited in 10 scientific papers (total in 10 papers)

Integration of equations of Kaup system kind with self-consistent source in class of periodic functions

A. B. Yakhshimuratova, B. A. Babajanovb

a Urgench Branch of Tashkent University of Information Technologies named after Muhammad al-Khwarizmi, Al-Khwarizmi street, 110, 220100, Urgench city, Uzbekistan
b Urgench State University, Hamid Alimjan street, 14, 220100, Urgench city, Uzbekistan
References:
Abstract: In this paper, we consider the equations of Kaup system kind with a self-consistent source in the class of periodic functions. We discuss the complete integrability of the considered nonlinear system of equations, which is based on the transformation to the spectral data of an associated quadratic pencil of Sturm–Liouville equations with periodic coefficients. In particular, Dubrovin-type equations are derived for the time-evolution of the spectral data corresponding to the solutions of equations of Kaup system kind with self-consistent source in the class of periodic functions. Moreover, it is shown that spectrum of the quadratic pencil of Sturm–Liouville equations with periodic coefficients associated with considering nonlinear system does not depend on time. In a one-gap case, we write the explicit formulae for solutions of the problem under consideration expressed in terms of the Jacobi elliptic functions. We show that if p0(x)p0(x) and q0(x)q0(x) are real analytical functions, the lengths of the gaps corresponding to these coefficients decrease exponentially. The gaps corresponding to the coefficients p(x,t)p(x,t) and q(x,t)q(x,t) are same. This implies that the solutions of considered problem p(x,t)p(x,t) and q(x,t)q(x,t) are real analytical functions in xx.
Keywords: equations of Kaup system kind, quadratic pencil of Sturm–Liouville equations, inverse spectral problem, trace formulas, periodical potential.
Funding agency Grant number
Keele University International Erasmus+Program KA106-2
The authors express their gratitude to Prof. Aknazar Khasanov (Samarkand State University, Uzbekistan) for discussion and valuable advice, as well as to the International Erasmus+Program KA106-2, Keele University, UK.
Received: 25.02.2019
Bibliographic databases:
Document Type: Article
UDC: 517.957
Language: English
Original paper language: English
Citation: A. B. Yakhshimuratov, B. A. Babajanov, “Integration of equations of Kaup system kind with self-consistent source in class of periodic functions”, Ufa Math. J., 12:1 (2020), 103–113
Citation in format AMSBIB
\Bibitem{YakBab20}
\by A.~B.~Yakhshimuratov, B.~A.~Babajanov
\paper Integration of equations of Kaup system kind with self-consistent source in class of periodic functions
\jour Ufa Math. J.
\yr 2020
\vol 12
\issue 1
\pages 103--113
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\crossref{https://doi.org/10.13108/2020-12-1-103}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85084251246}
Linking options:
  • https://www.mathnet.ru/eng/ufa506
  • https://doi.org/10.13108/2020-12-1-103
  • https://www.mathnet.ru/eng/ufa/v12/i1/p104
  • This publication is cited in the following 10 articles:
    1. B.A. Babajanov, Sh.O. Sadullaev, M.M. Ruzmetov, “Integration of the Kaup–Boussinesq system via inverse scattering method”, Partial Differential Equations in Applied Mathematics, 11 (2024), 100813  crossref
    2. B. A. Babajanov, F. B. Abdikarimov, F. U. Sulaymonov, “On the Integration of the Hierarchy of the Kaup–Boussinesq System with a Self-Consistent Source”, Lobachevskii J Math, 45:7 (2024), 3233  crossref
    3. B. A. Babajanov, A. Sh. Azamatov, R. B. Atajanova, “Integration of the Kaup–Boussinesq system with time-dependent coefficients”, Theoret. and Math. Phys., 216:1 (2023), 961–972  mathnet  crossref  crossref  mathscinet  adsnasa
    4. Bazar Babajanov, Michal Fečkan, Aygul Babadjanova, “On the Differential-Difference Sine-Gordon Equation with an Integral Type Source”, Mathematica Slovaca, 73:6 (2023), 1499  crossref
    5. B. A. Babajanov, A. K. Babadjanova, A. Sh. Azamatov, “Integration of the differential–difference sine-Gordon equation with a self-consistent source”, Theoret. and Math. Phys., 210:3 (2022), 327–336  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    6. Aknazar B. Khasanov, Bazar A. Babajanov, Dilshod O. Atajonov, “On the integration of the periodic Camassa–Holm equation with a self-consistent source”, Zhurn. SFU. Ser. Matem. i fiz., 15:6 (2022), 785–796  mathnet  crossref  mathscinet
    7. B. A. Babajanov, D. O. Atajonov, “On the integration of the periodical Camassa–Holm equation with an integral type source”, Russian Math. (Iz. VUZ), 66:11 (2022), 1–11  mathnet  crossref  crossref
    8. Bazar Babajanov, Fakhriddin Abdikarimov, “Exact Solutions of the Nonlinear Loaded Benjamin-Ono Equation”, WSEAS TRANSACTIONS ON MATHEMATICS, 21 (2022), 666  crossref
    9. B. A. Babajanov, M. M. Ruzmetov, “On the construction and integration of a hierarchy for the periodic Toda lattice with a self-consistent source”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 38 (2021), 3–18  mathnet  crossref
    10. A. Yakhshimuratov, T. Kriecherbauer, B. Babajanov, “On the Construction and Integration of a Hierarchy for the Kaup System with a Self-Consistent Source in the Class of Periodic Functions”, Z. mat. fiz. anal. geom., 17:2 (2021), 233  crossref
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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